The role of topological heterogeneity in the origin of extreme events in a network is investigated here. The dynamics of the oscillators associated with the nodes are assumed to be identical and influenced by mean-field repulsive interactions. An interplay of topological heterogeneity and the repulsive interaction between the dynamical units of the network triggers extreme events in the nodes when each node succumbs to such events for discretely different ranges of repulsive coupling. A high degree node is vulnerable to weaker repulsive interactions, while a low degree node is susceptible to stronger interactions. As a result, the formation of extreme events changes position with increasing strength of repulsive interaction from high to low degree nodes. Extreme events at any node are identified with the appearance of occasional large-amplitude events (amplitude of the temporal dynamics) that are larger than a threshold height and rare in occurrence, which we confirm by estimating the probability distribution of all events. Extreme events appear at any oscillator near the boundary of transition from rotation to libration at a critical value of the repulsive coupling strength. To explore the phenomenon, a paradigmatic second-order phase model is used to represent the dynamics of the oscillator associated with each node. We make an annealed network approximation to reduce our original model and, thereby, confirm the dual role of the repulsive interaction and the degree of a node in the origin of extreme events in any oscillator associated with a node.

Epilepsy is now considered a network disease that affects the brain across multiple levels of spatial and temporal scales. The paradigm shift from an epileptic focus—a discrete cortical area from which seizures originate—to a widespread epileptic network—spanning lobes and hemispheres—considerably advanced our understanding of epilepsy and continues to influence both research and clinical treatment of this multi-faceted high-impact neurological disorder. The epileptic network, however, is not static but evolves in time which requires novel approaches for an in-depth characterization. In this review, we discuss conceptual basics of network theory and critically examine state-of-the-art recording techniques and analysis tools used to assess and characterize a time-evolving human epileptic brain network. We give an account on current shortcomings and highlight potential developments towards an improved clinical management of epilepsy.

Transcutaneous auricular vagus nerve stimulation (taVNS) is a novel non-invasive treatment option for different diseases and symptoms, such as epilepsy or depression. Its mechanism of action, however, is still not fully understood. We investigated short-term taVNS-induced changes of local and global properties of EEG-derived, evolving functional brain networks from eighteen subjects who underwent two 1 h stimulation phases (morning and afternoon) during continuous EEG-recording. In the majority of subjects, taVNS induced measurable modifications of network properties. Network alterations induced by stimulation in the afternoon were clearly more pronounced than those induced by stimulation in the morning. Alterations mostly affected the networks’ topology and stability properties. On the local network scale, no clear-cut spatial stimulation-related patterns could be discerned. Our findings indicate that the possible impact of diurnal influences on taVNS-induced network modifications would need to be considered for future research and clinical studies of this non-pharmaceutical intervention approach.

Constructing networks from empirical time-series data is often faced with the as yet unsolved issue of how to avoid potentially superfluous network constituents. Such constituents can result, e.g., from spatial and temporal oversampling of the system’s dynamics, and neglecting them can lead to severe misinterpretations of network characteristics ranging from global to local scale. We derive a perturbation-based method to identify potentially superfluous network constituents that makes use of vertex and edge centrality concepts. We investigate the suitability of our approach through analyses of weighted small-world, scale-free, random, and complete networks.

We investigate topological and spectral properties of models of European and US-American power grids and of paradigmatic network models as well as their implications for the synchronization dynamics of phase oscillators with heterogeneous natural frequencies. We employ the complex-valued order parameter—a widely used indicator for phase ordering—to assess the synchronization dynamics and observe the order parameter to exhibit either constant or periodic or non-periodic, possibly chaotic temporal evolutions for a given coupling strength but depending on initial conditions and the systems’ disorder. Interestingly, both topological and spectral characteristics of the power grids point to a diminished capability of these networks to support a temporarily stable synchronization dynamics. We find non-trivial commonalities between the synchronization dynamics of oscillators on seemingly opposing topologies.

Background: The brain is never truly at rest. Even in the absence of external tasks, it remains active, continuously organizing itself into large-scale resting-state networks involved in shaping our internal thoughts and experiences. Understanding the networks’ structure and dynamics is key to uncovering how the brain functions as a whole. While previous studies have mapped resting-state networks and explored the roles of individual brain regions (network vertices), the relevance of the time-dependent functional interactions (network edges) between them remains largely unexplored. Methods: Here, we assess this relevance by elucidating the time-evolving importance of both brain regions and their interactions, associated with the networks’ constituents, using the fundamental concept of centrality. We investigate long-term electrophysiological recordings of brain dynamics from more than 100 participants and reveal new insights into how resting-state networks are organized over longer times. Results: Our findings reveal that the functional architecture of brain networks in a resting state is critically shaped by the dynamic interplay between brain regions. We identified functional importance backbones–core sets of dynamically central vertices and edges–whose configuration varies significantly between subgroups and further varies with different brain states, including wakefulness and sleep. Notably, regions associated with the default mode network exhibited adaptable patterns of centrality, challenging the notion of static network cores. Conclusions: By considering the temporal evolution of both vertices and edges, we provide a more comprehensive understanding of intrinsic brain activity and its functional relevance. This dynamic perspective reveals how the brain’s intrinsic activity is coordinated across space and time, highlighting the existence of functional importance backbones that adapt to different brain states.